Molecules in strong magnetic fields Trygve Helgaker, Kai Lange, Alessandro Soncini, and Erik Tellgren Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway

CTCC Department of Chemistry, University of Tromsø, Tromsø, Norway, December 17, 2008

Trygve Helgaker, Kai Lange, Alessandro Soncini, and Erik Tellgren Molecules (CTCC, in strong University magnetic of Oslo) fields

Molecules in Strong Magnetic Fields

1 / 29

Outline 1

Overview

2

Molecular magnetism

3

London orbitals

4

The LONDON program

5

The field-dependence of the energy of closed-shell molecules

6

An analytical model for paramagnetic closed-shell molecules

7

Molecular properties in strong magnetic fields

8

Potential-energy surfaces in strong magnetic fields

9

Conclusions

Helgaker et al. (CTCC, University of Oslo)

Overview

Molecules in Strong Magnetic Fields

2 / 29

Outline 1

Overview

2

Molecular magnetism

3

London orbitals

4

The LONDON program

5

The field-dependence of the energy of closed-shell molecules

6

An analytical model for paramagnetic closed-shell molecules

7

Molecular properties in strong magnetic fields

8

Potential-energy surfaces in strong magnetic fields

9

Conclusions

Helgaker et al. (CTCC, University of Oslo)

Overview

Molecules in Strong Magnetic Fields

2 / 29

Outline 1

Overview

2

Molecular magnetism

3

London orbitals

4

The LONDON program

5

The field-dependence of the energy of closed-shell molecules

6

An analytical model for paramagnetic closed-shell molecules

7

Molecular properties in strong magnetic fields

8

Potential-energy surfaces in strong magnetic fields

9

Conclusions

Helgaker et al. (CTCC, University of Oslo)

Overview

Molecules in Strong Magnetic Fields

2 / 29

Outline 1

Overview

2

Molecular magnetism

3

London orbitals

4

The LONDON program

5

The field-dependence of the energy of closed-shell molecules

6

An analytical model for paramagnetic closed-shell molecules

7

Molecular properties in strong magnetic fields

8

Potential-energy surfaces in strong magnetic fields

9

Conclusions

Helgaker et al. (CTCC, University of Oslo)

Overview

Molecules in Strong Magnetic Fields

2 / 29

Outline 1

Overview

2

Molecular magnetism

3

London orbitals

4

The LONDON program

5

The field-dependence of the energy of closed-shell molecules

6

An analytical model for paramagnetic closed-shell molecules

7

Molecular properties in strong magnetic fields

8

Potential-energy surfaces in strong magnetic fields

9

Conclusions

Helgaker et al. (CTCC, University of Oslo)

Overview

Molecules in Strong Magnetic Fields

2 / 29

Outline 1

Overview

2

Molecular magnetism

3

London orbitals

4

The LONDON program

5

The field-dependence of the energy of closed-shell molecules

6

An analytical model for paramagnetic closed-shell molecules

7

Molecular properties in strong magnetic fields

8

Potential-energy surfaces in strong magnetic fields

9

Conclusions

Helgaker et al. (CTCC, University of Oslo)

Overview

Molecules in Strong Magnetic Fields

2 / 29

Outline 1

Overview

2

Molecular magnetism

3

London orbitals

4

The LONDON program

5

The field-dependence of the energy of closed-shell molecules

6

An analytical model for paramagnetic closed-shell molecules

7

Molecular properties in strong magnetic fields

8

Potential-energy surfaces in strong magnetic fields

9

Conclusions

Helgaker et al. (CTCC, University of Oslo)

Overview

Molecules in Strong Magnetic Fields

2 / 29

Outline 1

Overview

2

Molecular magnetism

3

London orbitals

4

The LONDON program

5

The field-dependence of the energy of closed-shell molecules

6

An analytical model for paramagnetic closed-shell molecules

7

Molecular properties in strong magnetic fields

8

Potential-energy surfaces in strong magnetic fields

9

Conclusions

Helgaker et al. (CTCC, University of Oslo)

Overview

Molecules in Strong Magnetic Fields

2 / 29

Outline 1

Overview

2

Molecular magnetism

3

London orbitals

4

The LONDON program

5

The field-dependence of the energy of closed-shell molecules

6

An analytical model for paramagnetic closed-shell molecules

7

Molecular properties in strong magnetic fields

8

Potential-energy surfaces in strong magnetic fields

9

Conclusions

Helgaker et al. (CTCC, University of Oslo)

Overview

Molecules in Strong Magnetic Fields

2 / 29

Molecular para- and diamagnetism I When a magnetic field is applied to a molecule, one of two things can happen: I I

the energy is lowered: molecular paramagnetism the energy is raised: molecular diamagnetism

I Open-shell molecules are paramagnetic I I I

permanent magnetic moments (unpaired spins) the molecule reorients itself and moves into the field temperature dependent

I Closed-shell molecules are nearly all diamagnetic I I I I

induced magnetic dipole only—no permanent magnetic moment induced currents oppose the external field (Lenz’ law) temperature independent much weaker than open-shell paramagnetism

I Some closed-shell molecules are paramagnetic

induced magnetic dipole only temperature independent I much weaker than the temperature-dependent open-shell paramagnetism I first discovered for MnO− (1914) 4 I much studied: BH and CH+ I General theory ov Van Vleck (1932) I I

Helgaker et al. (CTCC, University of Oslo)

Molecular magnetism

Molecules in Strong Magnetic Fields

3 / 29

Electronic Hamiltonian in an external magnetic field B I The external magnetic field is represented by a vector potential

B(r) = ∇ × A(r),

A(r) = 21 B × r

I The non-relativistic electronic Hamiltonian (atomic units)

H = H0 + A (r) · p + B (r) · s + 12 A (r)2 = H0 + 21 B · L + B · s + 18 (B × r) · (B × r) I I I I

H0 is the field-free non-relativistic Hamiltonian p = −i∇ is the generalized momentum operator L = r × p is the orbital angular momentum operator s = σ/2 is the spin angular momemtum operator

I The Hamiltonian depends both linearly and quadratically on the field I I

the linear term may lower or raise the energy the quadratic term will always raise the energy

I We can expect a rather complicated dependence of the energy on the field I I I

for large B, the second-order term will dominate but what happens for small and intermediate B? we consider only closed-shell states

Helgaker et al. (CTCC, University of Oslo)

Molecular magnetism

Molecules in Strong Magnetic Fields

4 / 29

Perturbation theory I Magnetic interactions are usually studied by perturbation theory

E (B) = E (0) −

X α

µα B −

1 2

X αβ

χαβ Bα Bβ + · · ·

I The first-order term represents interaction with the permanent magnetic moment

˛ ¸ ˙ ˛ µ = − 0 ˛ 12 L + s˛ 0 I

temperature-dependent paramagnetism (vanishes for closed-shell systems)

I The second-order term represents interaction with the induced dipole moment

˛ E D ˛ ˛ ˛ χ = − 14 0 ˛rrT − (rT r)I3 ˛ 0 +

1 2

˙ ˛ ˛ ¸ X h0 |L| ni n ˛LT ˛ 0 n

En − E0

I

the Langevin term arises from precessional motion of the electrons I temperature-independent diamagnetism

I

the sum-over-states term arises from orbital unquenching I temperature-independent paramagnetism

Helgaker et al. (CTCC, University of Oslo)

Molecular magnetism

Molecules in Strong Magnetic Fields

5 / 29

The need for a nonperturbative treatment I It is possible to go to higher orders by including hypermagnetizabilities

E (B) = E (0) −

X

1 2

αβ

χαβ Bα Bβ −

1 24

X αβγδ

Xαβγδ Bα Bβ Bγ Bδ + . . .

I However, the field dependence on the energy can be very complicated æ

æ

-756.680 æ

æ æ

æ

-756.685

æ

æ

æ

æ æ

æ

-756.690

æ

æ

æ

æ æ

æ

-756.695

æ æ æ

æ æ æ

-756.700 æ æ æ æ -756.705 æ æ æ æ æ æ æ æ -756.710 æ æ æ æ æ æ æ æ ææææ æ æ ææææ ææ æ æ ææ ææ æ æ ææ ææ æææ ææææææææææææ ææææææææææ

-0.04

I I

-0.02

0.02

0.04

the energy of C20 (ring conformation) as a function of B (atomic units) Taylor expansions useless for strong fields

I How does this complicated field dependence arise? Helgaker et al. (CTCC, University of Oslo)

Molecular magnetism

Molecules in Strong Magnetic Fields

6 / 29

Nonperturbative treatment of molecules in strong fields I Most studies of molecular magnetism are based on low-order expansions I I

magnetizabilities and hypermagnetizabilities such studies provide much useful information

I However, expansions around zero field have many limitations I I

the behaviour in strong fields cannot be studied many phenomena may remain unnoticed

I We have undertaken a nonperturbative study of molecules in strong fields I I I

this required the development of a new code complex orbitals and complex wave functions London atomic orbitals to remove gauge-origin dependence

I In the remainder of the talk, we will discuss the following points I I I I I

the need for London orbitals London-orbital integral evaluation in strong fields the energy in strong magnetic fields two-level analytic model for strong fields molecular properties in strong fields

Helgaker et al. (CTCC, University of Oslo)

Molecular magnetism

Molecules in Strong Magnetic Fields

7 / 29

Gauge-origin dependence and London orbitals I A uniform external field may be represented by any potential of the form

AO (r) = 12 B × (r − O) I I

the vector potential vanishes as the gauge origin O the position of the origin is not unique

I In exact theory, this non-uniqueness does not matter I

a change in the origin represents a gauge transformation the exact wave function undergoes a corresponding gauge transformation

I

all choices of gauge origin O then lead to the same (observable) results

I

ΨO = exp[iAK (O) · r]ΨK I In approximate calculations, our results in general do depend on the origin I

approx. wave functions are not sufficiently flexible to be properly gauge transformed

I This problem is solved by using gauge transforming the individual atomic orbitals (AOs) I

each AO has a unique “best” or “favoured” gauge origin: its atomic center ωlm = exp[iAK (O) · r]χlm (rK ) ← gauge transformation from AO to global origin

I I

each AO behaves as if the gauge origin were at its center the use of such London orbitals removes the gauge-origin dependence

Helgaker et al. (CTCC, University of Oslo)

London orbitals

Molecules in Strong Magnetic Fields

8 / 29

The efficacy of London orbitals I London orbitals are AOs with an attached complex phase factor

ωlm = exp[iAK (O) · r]χlm (rK ) I

gauge factor removes gauge-origin dependence of magnetic properties

I London orbitals are correct to first-order in the external magnetic field I

for this reason, basis-set convergence is usually improved

I Calculations on the water molecule

STO-3G cc-pVDZ cc-pVTZ aug-cc-pVDZ I I

Lon −2.6 −2.8 −2.9 −3.0

χzz CM −2.7 −2.8 −2.9 −3.0

H −6.3 −4.0 −3.2 −3.3

Lon 3.6 5.7 7.9 15.5

Xzzzz CM 1.0 5.5 9.4 16.1

H −0.6 6.1 10.5 15.9

London orbitals greatly improve convergence of magnetizabilities they are less efficacious for hypermagnetizabilities

Helgaker et al. (CTCC, University of Oslo)

London orbitals

Molecules in Strong Magnetic Fields

9 / 29

Hybrid plane-wave–Gaussians (PWG) orbitals I London AOs are in fact hybrid plane-wave–Gaussian (PWG) orbitals

ωκ,c (r) = exp(iκ · r) Slm (r) exp(−arA2 ) | {z } | {z }

plane wave solid-harmonic Gaussian

I

the wave vector κ is the AO-centered vector potential at the gauge origin

I More generally, PWGs have several uses: I I

mixed basis for periodic boundary conditions and scattering studies gauge-origin independent magnetic properties at zero field

I Requires a generalization of GTO integral-evaluation techniques I I

at zero field, complex algebra may be avoided at finite field, complex algebra cannot be avoided

I We have developed and implemented a McMurchie–Davidson PWG scheme I I

Tellgren et al., JCP 129, 154114 (2008) previous work: M. Tachikawa and M. Shiga, Phys. Rev. E 64:056706 (2001)

Helgaker et al. (CTCC, University of Oslo)

London orbitals

Molecules in Strong Magnetic Fields

10 / 29

PWG product rule and overlap integrals I

In many respects, a straightforward generalization of GTO integral evaluation

I

The Gaussian product rule still holds 2 ∗ 2 Ωκλ ab (r) = exp(iκ · r) exp(−arA ) exp(iλ · r) exp(−brB ) {z }| {z } |

PWG at A

PWG at B

2 ab = exp(− a+b RAB ) exp[−i(κ − λ) · r] exp[−(a + b)rP2 ] {z } | {z }|

prefactor

I

PWG at P = (aA + bB)/(a + b)

Integration over all space yields Z

    π 3/2 exp − ab R 2 2 a+b AB (κ−λ) Ωκλ ab (r) dr = exp − 4(a+b) + i(κ − λ) · P (a + b)3/2 | {z }| {z } plane-wave contribution

Helgaker et al. (CTCC, University of Oslo)

London orbitals

standard Gaussian overlap

Molecules in Strong Magnetic Fields

11 / 29

PWG two-electron integrals I

As for standard Gaussians, Coulomb integrals reduce to the Boys function 2 2 exp(iκ · r1 ) exp(−pr1P ) exp(iλ · r2 ) exp(−qr2Q ) dr1 dr2 r12 h 2 i 2π 5/2 h i 2 2 pq κ √ = exp − 4p − λ4q − iκ · P − iλ · Q F0 p+q (P0 − Q0 ) pq p + q | {z }

ZZ

J=

P0 = P − iκ/2p, Q0 = Q − iλ/2q

I

The Boys function is given by Z F0 (x) =

1

exp(−xt 2 ) dt

← complex argument x

0 I

I

evaluated in the usual manner by expansion and recursion

For functions of higher angular momentum, recurrence relations are used I I

some translational symmetry lost more complicated recurrence relations

Helgaker et al. (CTCC, University of Oslo)

London orbitals

Molecules in Strong Magnetic Fields

12 / 29

The LONDON program I I

an ab initio program for finite-field calculations with London orbitals some features of the present code I I I I

I

some restrictions of the present code I I

I

Hartree–Fock theory implemented recent Kohn–Sham implementation many first-order properties (x − Cx )m (∂/∂x)n excitation energies using response theory only uncontracted Cartesian basis functions closed-shell wave functions only

code written by Erik Tellgren and Alessandro Soncini I I I

mostly C++, some Fortran 77 modular but not highly optimized yet C20 is a “large” system

Helgaker et al. (CTCC, University of Oslo)

The LONDON program

Molecules in Strong Magnetic Fields

13 / 29

The dependence of total energy on the magnetic field I RHF calculations of154114-8 B dependence for different systems Tellgren, Soncini, and Helgaker a)

b)

J. Chem. Phys. 129, 154114 !2008"

−3

14

x 10

0.1 12 0.08

10 8

0.06

6 0.04 4 0.02

0 −0.1

2

−0.05

0

0.05

0 −0.1

0.1

c)

−0.05

0

0.05

0.1

d) 0

−0.002

0.02

−0.004 0.01

−0.006 −0.008

0

−0.01 −0.01

−0.012 −0.014

−0.02

−0.016 −0.018

−0.03

−0.02 −0.1

a b c d

−0.05

0

0.05

0.1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

FIG. 1. Energy as a function of the magnetic field for different systems. Triangles represent results from finite-field calculations and solid lines are quartic fitting polynomials. !a" Benzene !with the aug-cc-pVDZ basis" illustrates the typical case of diamagnetic quadratic dependence in response to an out-of-plane field. !b" Cyclobutadiene !aug-cc-pVDZ" deviates from the typical case by exhibiting a nonquadratic dependence on an out-of-plane field. !c" Boron monohydride !aug-cc-pVTZ" is an interesting case of positive magnetizability for a perpendicular field, exhibiting nonquadratic behavior. !d" Boronmonohydride !aug-cc-pVTZ" in a larger range of perpendicular fields, exhibiting a clearly nonperturbative behavior.

benzene (aug-cc-pVDZ): typical case of diamagnetic quadratic dependence cyclobutadiene (aug-cc-pVDZ): non-quadratic dependence on an out-of-plane field For the perpendicular components, the estimates of the linear magnetizability are in fact positive and large enough to hypermagnetizability we obtain using the field above mentioned make evenparamagnetic the average magnetizabilitydependence positive !paramagBH (aug-cc-pVTZ): for a perpendicular fitting procedure are not robust, varying with the number of netic". It is therefore interesting to verify via our finite-field data points included in thereveals least-squares fitting and the deLondon-orbital approachrange whether thisof verythe small system is BH (aug-cc-pVTZ): larger perpendicular field nonperturbative behaviour indeed characterized by a particularly large nonlinear magnetic response. The geometry used for the calculations is that optimized at the multiconfigurational SCF level in Ref. 51,

gree of the polynomial. Using 41 uniformly spaced field values in the range −0.1– 0.1 a.u. and a fitting polynomial of order 16, we arrive at reasonably converged values of !! Molecules in Strong a.u. for the magnetizability and

Helgaker et al. (CTCC, University of corresponding Oslo) The field-dependence of closed-shell = 7.1 a.u. and energies X! = −8 " 103 to a bond length of rBH = 1.2352 Å.

Magnetic Fields

14 / 29

The field dependence of paramagnetic molecules I

We have studied the field-dependence of a number of molecules O

B

H

C

2

C4H4

C8H8

5 C16H10

7

I

O

1

4

Mn

H

O O

3 C12H12

6 C20H10

8

For all these systems, the field dependence takes on a sombrero shape I I

we will give some examples we will explain this behaviour by means of a simple analytical model

Helgaker et al. (CTCC, University of Oslo)

The field-dependence of closed-shell energies

Molecules in Strong Magnetic Fields

15 / 29

0.1 W!W0 !au"

Small paramagnetic molecules a)

0.05

0.1

0.15

0.2

I

0.3

0.3

0.4

0.5

0.6

STO3!G, Bc " 0.43

! 0.04

Bc " 0.44

DZ,

aug!DZ, Bc " 0.45

! 0.06

+ , and MnO− Three closed-shell paramagnetic molecules: BH, !CH 0.08 4 ! 0.02

I I

a)

field applied perpendicularly for BH and CH+ !0.1 aug-cc-pVDZ on all atoms except Wachters-f for Mn ! 0.12 ! 0.03

! 0.04

W!W0 !au"

B !au"

BH

0.05

0.1

0.15

b)

0.2

0.25

0.3

STO!3G, Bc " 0.24 DZ, Bc " 0.22 aug!DZ, Bc " 0.23

! 0.01

! 0.02

! 0.04

b)

W!W0 !au"

CH# 0.1

0.2

0.3

0.4

0.5

0.6

B !au"

c)

! 0.02

STO3!G, Bc " 0.43

! 0.04

Bc " 0.44

DZ,

aug!DZ, Bc " 0.45

! 0.06

! 0.03

I

0.25

STO!3G, Bc " 0.24 DZ, Bc " 0.22 aug!DZ, Bc " 0.23

! 0.01

0.2

! 0.02

B !au"

BH

W!W0 !au" 0.1

MnO4! 0.2

0.3

0.4

0.5

0.6

0.7

B !au"

STO!3G, Bc " 0.45 Wachters, Bc " 0.50

! 0.1 ! 0.2

! 0.08

! 0.3

! 0.1

! 0.4

! 0.12

W!W0 !au"

CH#

B !auc) "

W!W0 !au"

MnO4!

B !au"

Energy minimum occurs at a characteristicB critical field Bc 0.1

0.2

0.3

0.4

0.5

0.6

! 0.1

! 0.02

I

! 0.04

I

! 0.06

I

! 0.08

0.1

0.2

0.3

0.4

0.5

0.6

0.7

STO!3G, c " 0.45 Wachters, Bc " 0.50

G, B 0.43 we may in STO3 principle separate molecules by applying a field gradient ! 0.2 DZ, B 0.44 Bc = 0.2–0.5 these small systems aug DZ,a.u. B 0.45 for ! 0.3 ! 0.4 strongest fields attainable: 100 T (4.3 · 10−4 a.u.) !

!

c

"

c

"

c

"

! 0.1 ! 0.12

c)

W!W0 !au" 0.1 ! 0.1

MnO4! 0.2

0.3

0.4

0.5

0.6

0.7

B !au"

STO!3G, Bc " 0.45 Wachters, Bc " 0.50

! 0.2 ! 0.3

Helgaker et al. (CTCC, University of Oslo)

The field-dependence of closed-shell energies

Molecules in Strong Magnetic Fields

16 / 29

6!31G cc! pVDZ

0.025

! 0.002

0.02

Antiaromatic closed-shell [4n]-carbocycles ! 0.003

0.015 0.01

I

! 0.004

Their linear response is characterized by strong ring currents 0.005

I

a) I

B !au"

! 0.005

cyclobutadiene C4 H4 , cyclo-octatetraene C8 H8 and [12]-annulene C12 H12 b) magnetic field along principal axis 0.02

W!W0 !au"

C4H4: total energy

0.03

0.08

0.1

0.12

0.08

0.1

0.12 W!W0 !au"

C12H12: total energy

STO!3G, Bc "0.002 0.12 6!31G, Bc " 0.11 cc!pVDZ, Bc " 0.10

0.004

0.0015

STO!3G, Bc " 0.035 6!31G, Bc " 0.034 cc!pVDZ, Bc " 0.032

! 0.002

0.02

d)

C4H4: #!energy 0.02 C8H0.04 0.06 8: total energy

! 0.001

STO!3G 6!31G cc! pVDZ

0.025

0.06

W!W0 !au"

c)

W! W0 !au"

0.04

STO!3G, Bc " 0.018 6!31G, Bc " 0.018 cc!pVDZ, Bc " 0.016

0.001

0.002

0.01

! 0.004

0.01

0.02

0.03

0.04

0.05

0.005

0.06

0.01

0.015

0.02

0.025

0.03

B !au"

! 0.0005

0.005

! 0.002

0.02

I

0.0005

! 0.003

0.015

0.04

0.06

0.08

0.1

0.12

B !au"

! 0.005

! 0.001

Thec) critical field is now one d) order of magnitude smaller

B!auc " ≈ 0.032 for forC C H12 !W !au" C H : total energyC8 H8 ; Bc ≈W0.016 H :12 total energy 0.002 the critical field decreases with increasing size of the system I 0.0015 of the area 0.004Bc should vary as the inverse STO!3G, of B " 0.018 the molecule STO!3G, B " 0.035 6!31G, B " 0.018 6!31G, B " 0.034 I we estimate that B should0.001 cc!pVDZ, B for " 0.016C72 H72 be observable c cc!pVDZ, B " 0.032

IW!W

0

8

0

8

12

12

I

c

c

c

c

c

c

0.002 0.0005

0.01

0.02

0.03

0.04

0.05

0.005

0.06

0.01

0.015

0.02

0.025

0.03

B !au"

! 0.0005

! 0.002

Helgaker et al. (CTCC, University of Oslo)

! 0.001

The field-dependence of closed-shell energies

Molecules in Strong Magnetic Fields

17 / 29

BH energy in a perpendicular magnetic field 154114-9 Calculations in strong magnetic fields using London orbitals I Polynomial fits to the BH energy in perpendicular magnetic field

nents of hy polynomial equately. U we can affo rate estimat netizability tion betwe $-electrons netizability linear magn on 41 un −0.1– 0.1 a converged t

data points degree 6 degree 10 degree 14

−25.135 −25.14 −25.145 −25.15 −25.155 −25.16 −25.165 −25.17 0 I I I Helgaker et al.

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

FIG. 2. !Color online" Boron monohydride !aug-cc-pVTZ" energies for a

HF/aug-cc-pVDZ level of theory large range of perpendicular magnetic fields together with least-squares fitonly even-order terms are included by symmetry ted polynomials ofgreater different expansions of order 14 or aredegrees. needed The fitting polynomials contain only

IV. CONCL

We

h

even powers due to time reversal symmetry #W!H" = W!−H"$. In this range McMurchie (CTCC, University of Oslo) The field-dependence of closed-shell energies Molecules in Strong Fields 18 / 29 of fields, polynomials of very high degree are required in order to Magnetic ad-

Analytical model for the diamagnetic transition I

Molecular orbitals relevant for BH: 1sB , 2σBH , 2px , 2py , 2pz

I

Ground and excited states: 2 |0i = |1sB2 2σBH 2pz2 |,

I

(molecule along z axis)

2 |xi = |1sB2 2σBH 2pz 2px |,

2 |y i = |1sB2 2σBH 2pz 2py |

Let us apply a perpendicular magnetic field in the y direction: HB = 12 BLy + 18 B 2 (x 2 + z 2 )

I

This leads to the following Hamiltonian matrix:    h0|H|0i h0|H|y i E0 − 21 χ0 B 2 H(B) = = hy |H|0i hy |H|y i −iµB

iµB E1 − 12 χ1 B 2



where we have introduced χ0 = 14 h0|x 2 + z 2 |0i, Helgaker et al. (CTCC, University of Oslo)

χ1 = 14 hy |x 2 + z 2 |y i,

Two-level model for paramagnetic molecules

µ = − 21 ih0|Ly |y i Molecules in Strong Magnetic Fields

19 / 29

Energy levels of the two-level model I Two-level Hamiltonian:

„ H(B) =

−∆ − 12 χ0 B 2 −iµB

iµB ∆ − 21 χ1 B 2

«

„ =

−0.01 + 7B 2 −iµB

iµB 0.01 + 4B 2

«

I Eigenvalues:

1 1 W0/1 (B) = − (χ0 + χ1 )B 2 ∓ 2 2

q

[2∆ + (χ0 − χ1 )B 2 ]2 + 4µ2 B 2

I Plots for different values of µ: I

uncoupled (0), diamagnetic (0.2), nonmagnetic (0.374), and paramagnetic (0.6) 0.09

0.09

0.06

0.06

0.03

0.03

-0.1

0.1

-0.1 -0.03

0.09

0.09

0.06

0.06

0.03

0.03

-0.1

0.1

-0.1

-0.03

I

0.1

-0.03

0.1 -0.03

magnetizability (negative second derivative at B = 0): χ0 + µ2 /2∆ = −7, −5, 0, 11

Helgaker et al. (CTCC, University of Oslo)

Two-level model for paramagnetic molecules

Molecules in Strong Magnetic Fields

20 / 29

Example 2: A two-state model

Two-level model fitted to experimental data I

The two-level model contains four parameters I I

Setting ∆E = 0.097 (zero field excitation energy) and fi yields:

may be fitted to experimental data parameters tosuperior the ground energies provides excellent fits, to polynomial fits BH singlet energies (aug!cc!pVDZ)

0.25

Energy E(Bx)

0.2

0.15

0.1

g.s. exc!1 exc!2 fitted fitted

0.05

0

!0.05 0 Helgaker et al. (CTCC, University of Oslo)

0.05

0.1

0.15

0.2 0.25 Field Bx

0.3

Two-level model for paramagnetic molecules

0.35

0.4

0.45

Molecules in Strong Magnetic Fields

21 / 29

% 0.02

Two-level model fitted to experimental data % 0.03

I

The two-level model contains four parameters I I I

may be fitted to experimental data % 0.04 provides excellent fits, superior to polynomial fits comparisons with 6- and 8-order polynomial fits BH: Polynomial and two%level model fits

a)

0.1

0.2

0.3

c)B

0.4

b)

DIAMAGNETIC

% 0.02 % 0.01

0.2

% 0.04

% 0.03

0.15

PARAMAGNETIC

Bc

% 0.06

% 0.02

% 0.08

0.1

% 0.1

0.05

% 0.12

0

% 0.04

c)

d)

BH model fits CH$ : Polynomial and two%level B 0.25 0.1 0.2 0.3 0.4 0.5 0.6

0

CH$ : Polynomial and two%level model fits 0.1

0.2

0.3

% 0.02 % 0.04 Helgaker et al. (CTCC, University of Oslo)

0.4

0.5

0.6

B

e)

0.2

0.4 0.6 2 !! "0!"#2

0.8

1

CH$

d) 2%

C16H10 : Polynomial and two%level model fits B DIAMAGNETIC 0.4 0.005 0.01 0.015 0.02 0.025 0.03

Two-level model for paramagnetic 0.3 % 0.001 molecules

Molecules in Strong Magnetic Fields

f)

22 / 29

C20 : more structure æ

æ

-756.680 æ

æ æ

æ

-756.685

æ

æ

æ

æ æ

æ

-756.690

æ

æ

æ

æ æ

æ

-756.695

æ æ æ

æ æ æ

-756.700 æ æ æ æ -756.705 æ æ æ æ æ æ æ æ -756.710 æ æ ææææææææææ æ æ ææææ ææ æ æ ææ ææ æ æ ææ ææ æææ æ æ æ ææææææææææ æææææææææ

-0.04

Helgaker et al. (CTCC, University of Oslo)

-0.02

0.02

Molecular properties in strong magnetic fields

0.04

Molecules in Strong Magnetic Fields

23 / 29

Induced magnetic moment and angular momentum I The induced magnetic moment M and angular momentum L are related as

M = − 12 L = − 21 E 0 (B),

∆E = −MB

I Diamagnetic molecules: I

M is always aligned against the field, increasing the energy

I Paramagnetic molecules: I I I I

M M M M

first aligns with the field, decreasing the energy reaches its maximum value at the inflection point E 00 (B) = 0 Example 2: Non-perturbative phenomena then decreases again until it vanishes at Bc then aligns against the field, making the system diamagnetic

BH properties (aug-cc-pVDZ) as function of perpendicular field: Energy

Angular momentum

Nuclear shielding integral

1

!25.11

Boron Hydrogen

0.4

| L / |r ! C

nuc

0.5

!25.13

Lx(Bx)

E(Bx)

!25.12

!25.14

0

x

0

0.2

!25.15 !0.2 !25.16 !0.5 0

0.2

0.4

0

Orbital energies

I

0.2

0.4

0

HOMO!LUMO gap

0.2

0.4

Singlet excitation energies

0.5 There is 0.1 no net induced angular momentum at the0.4energy minimum 0.48

0 of Oslo) Helgaker et al. (CTCC, University

Molecular properties in strong magnetic fields

0.35

Molecules in Strong Magnetic Fields

24 / 29

Lx / |r ! Cn

Lx(Bx)

E(Bx)

0.5

!25.13

0.2

Oribtal and excitation energies in a magnetic field !25.14

0

0

!25.15 !0.2

I We!25.16 have implemented linear response theory in finite magnetic fields !0.5

I BH orbital gap and 0.2 singlet excitations 0 energies, 0.2 HOMO–LUMO 0.4 0 0.4 0 Orbital energies

HOMO!LUMO gap

0.2

0.4

Singlet excitation energies

0.5 0.4

0.1 0.48

LUMO HOMO

!0.1

0.35

0.46

"(Bx)

!gap(Bx)

!(Bx)

0

0.44 0.42

!0.2

0.2

0.4

!0.3

0.15

0.38 0

I I I

0.2

0.4

0.3 0.25

0

0.2

0.4

0.1

0

0.2

0.4

the HOMO and LUMO orbital energies decrease and increase, respectively, with B the HOMO–LUMO gap opens up with increasing B most excitation energies increase with increasing B

Helgaker et al. (CTCC, University of Oslo)

Molecular properties in strong magnetic fields

Molecules in Strong Magnetic Fields

25 / 29

H2 potential-energy curve in a perpendicular magnetic field I The magnetic field changes the shape of the potential-energy curve

-0.5

-0.6

-0.7

-0.8 B=0.45 -0.9

B=0.30 B=0.15

-1.0 B=0.00

-1.1

1.0

I I

1.5

2.0

2.5

3.0

diamagnetic behaviour at all separations most pronounced for atoms (no paramagnetic term)

I The bond length of H2 decreases with increasing magnetic field Helgaker et al. (CTCC, University of Oslo)

Potential-energy surfaces in strong magnetic fields

Molecules in Strong Magnetic Fields

26 / 29

F2 potential-energy curve in a perpendicular magnetic field I The magnetic field changes the shape of the potential-energy curve

-198.4 B=0.00 B=0.45 B=0.05 -198.5

B=0.10

B=0.30 -198.6

-198.7 2.5

I I

3.0

3.5

4.0

4.5

5.0

5.5

diamagnetic behaviour in the molecular limit paramagnetic behaviour in the atomic limit

I The bond length of F2 increases with increasing magnetic field Helgaker et al. (CTCC, University of Oslo)

Potential-energy surfaces in strong magnetic fields

Molecules in Strong Magnetic Fields

27 / 29

BH potential-energy curve in a perpendicular magnetic field I The magnetic field changes the shape of the potential-energy curve

-24.4

-24.6

-24.8

B=0.45

B=0.30 B=0.00 -25.0

B=0.15

-25.2 2

I I

3

4

5

mixed para- and diamagnetic behaviour at all separations mostly diamagnetic in the atomic limit

I The bond length of BH decreases with increasing magnetic field Helgaker et al. (CTCC, University of Oslo)

Potential-energy surfaces in strong magnetic fields

Molecules in Strong Magnetic Fields

28 / 29

Conclusions I

We have developed the LONDON program I I I I I

I

The LONDON program may be used for I I

I

finite-difference alternative to analytical derivatives studies of molecules in strong magnetic fields

We have studied the behaviour of paramagnetic molecules in strong fields I I

I

complex orbitals and wave functions restricted Hartree–Fock and Kohn–Sham theories London atomic orbitals for gauge-origin independence expectation values of one-electron operators linear response theory

all paramagnetic molecules attain a global minimum at a characteristic field Bc Bc decreases with system size and should be observable for C72 H72

This behaviour can be understood from a simple two-level model I I

explains the existence of a global minimum at Bc at Bc , the induced angular momentum vanishes

Helgaker et al. (CTCC, University of Oslo)

Conclusions

Molecules in Strong Magnetic Fields

29 / 29